Vardaan Learning Institute
Chapter 1 Notes
Orienting Yourself: The Use of Coordinates
1. Need for a Coordinate System
To pinpoint the exact location of an object or a point on a flat surface (a plane), a single number is not sufficient. We require a reference system using two perpendicular lines.
Core Definition
Cartesian Coordinate System: A system invented by René Descartes that uses a pair of perpendicular axes to determine the position of any point on a plane.
2. Elements of the Cartesian Plane
- X-axis: The horizontal number line $X'OX$.
- Y-axis: The vertical number line $Y'OY$.
- Origin ($O$): The point of intersection of the X-axis and Y-axis, represented as $(0, 0)$.
- Quadrants: The two axes divide the plane into four equal regions called Quadrants (numbered I, II, III, and IV counter-clockwise).
| Quadrant |
X-coordinate (Abscissa) |
Y-coordinate (Ordinate) |
Sign Pair $(x, y)$ |
Example Point |
| Quadrant I |
Positive ($> 0$) |
Positive ($> 0$) |
$(+, +)$ |
$(3, 5)$ |
| Quadrant II |
Negative ($< 0$) |
Positive ($> 0$) |
$(-, +)$ |
$(-4, 2)$ |
| Quadrant III |
Negative ($< 0$) |
Negative ($< 0$) |
$(-, -)$ |
$(-2, -6)$ |
| Quadrant IV |
Positive ($> 0$) |
Negative ($< 0$) |
$(+, -)$ |
$(5, -1)$ |
3. Coordinates of a Point
Coordinates of a point $P = (\text{Abscissa}, \text{Ordinate}) = (x, y)$
- Abscissa ($x$-coordinate): The perpendicular distance of a point from the $Y$-axis.
- Ordinate ($y$-coordinate): The perpendicular distance of a point from the $X$-axis.
- Any point on the $X$-axis has an ordinate of $0$, so its coordinates are of the form $(x, 0)$.
- Any point on the $Y$-axis has an abscissa of $0$, so its coordinates are of the form $(0, y)$.
4. Solved Examples
Example 1: Identifying Quadrants & Axes
Question: State the quadrant or axis on which each of the following points lies: $A(4, -3)$, $B(-2, 0)$, $C(-5, -4)$, $D(0, 7)$.
Solution:
- Point $A(4, -3)$: $x > 0$ and $y < 0 \implies$ Quadrant IV.
- Point $B(-2, 0)$: $y = 0 \implies$ Lies on the negative X-axis.
- Point $C(-5, -4)$: $x < 0$ and $y < 0 \implies$ Quadrant III.
- Point $D(0, 7)$: $x = 0 \implies$ Lies on the positive Y-axis.
Example 2: Perpendicular Distance
Question: Find the perpendicular distances of the point $P(-6, 8)$ from both axes.
Solution:
- Distance from $Y$-axis $= |x\text{-coordinate}| = |-6| = 6$ units.
- Distance from $X$-axis $= |y\text{-coordinate}| = |8| = 8$ units.
5. Chapter Summary
Quick Recap
- The origin is $(0,0)$.
- Distance is always a non-negative quantity.
- Order matters: $(x, y) \neq (y, x)$ unless $x = y$.
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